2. PRESENTATION BY
OSAMA TAHIR
09-EE-88

4. COMPLEX NUMBERS
A complex number is a number consisting
of a Real and Imaginary part.
It can be written in the form
i  1

5. COMPLEX NUMBERS
 Why complex numbers are introduced???
Equations like x2=-1 do not have a solution within
the real numbers
x  1
2
x  1
i  1
i  1
2

6. COMPLEX CONJUGATE
 The COMPLEX CONJUGATE of a complex number
z = x + iy, denoted by z* , is given by
z* = x – iy
 The Modulus or absolute value
is defined by
z x y 2 2

7. COMPLEX NUMBERS
Real numbers and imaginary numbers are
subsets of the set of complex numbers.
Real Numbers Imaginary
Numbers
Complex Numbers

8. COMPLEX NUMBERS
Equal complex numbers
Two complex numbers are equal if their
real parts are equal and their imaginary
parts are equal.
If a + bi = c + di,
then a = c and b = d

9. ADDITION OF COMPLEX NUMBERS
(a  bi)  (c  di)  (a  c)  (b  d )i
Imaginary Axis
z2
EXAMPLE z sum
z1
(2  3i )  (1  5i )
z2
 (2  1)  (3  5)i Real Axis
 3  8i

10. SUBTRACTION OF COMPLEX
NUMBERS
(a  bi)  (c  di)  (a  c)  (b  d )i
Imaginary Axis
Example z1 z 2
( 2  3i )  (1  5i ) z2
z diff
 ( 2  1)  (3  5)i z 2
Real Axis
 1  2i

11. MULTIPLICATION OF COMPLEX
NUMBERS
(a  bi)(c  di)  (ac  bd)  (ad  bc)i
Example
( 2  3i )(1  5i )
 ( 2  15)  (10  3)i
 13  13i

12. DIVISION OF A COMPLEX
NUMBERS
a  bi 
a  bi  c  di
c  di c  di c  di
ac  adi  bci  bdi 2

c d
2 2
ac  bd  bc  ad i

c d
2 2

13. EXAMPLE
6  7i  6  7i   1  2i 

1  2i  1  2i  1  2i 
6  12i  7i  14i 2
6  14  5i
 
12  2 2 1 4
20  5i 20 5i
    4i
5 5 5

14. COMPLEX PLANE
A complex number can be plotted on a plane with two
perpendicular coordinate axes
 The horizontal x-axis, called the real axis
 The vertical y-axis, called the imaginary axis
y
P
x-y plane is known as the
z = x + iy complex plane.
O x
The complex plane
Slide 14

15. COMPLEX PLANE
Geometrically, |z| is the distance of the point z from the
origin while θ is the directed angle from the positive x-
axis to OP in the above figure.
y
  t an1 
 
x
θ is called the argument of z and is
denoted by arg z. Thus,
Im
P
 y y
  arg z  tan  
1
z0 z = x + iy
 x r
|z |=
θ
O x Re

16. Expressing Complex Number
in Polar Form
x  r cos  y  r sin 
So any complex number, x + iy,
can be written in
polar form:
x  yi  r cos  r sin i

17. Imaginary axis
Real axis

18. De Moivre’s Theorem
De Moivre's Theorem is the theorem which
shows us how to take complex numbers to any
power easily.
Let r(cos F+isin F) be a complex number
and n be any real number. Then
[r(cos F+isin F]n = rn(cosnF+isin nF)
[r(cos F+isin F]n = rn(cosnF+isin nF)

19. Euler Formula
The polar form of a complex number can be rewritten as
z  r (cos  j sin  )  x  jy
 re j
This leads to the complex exponential
function :
z  x  jy
e z  e x  jy  e x e jy
 e x cos y  j sin y 

20. Example
A complex number, z = 1 - j
has a magnitude
| z | (12  12 )  2
 1   
and argument : z  tan    2n     2n  rad
1
 1   4 

Hence its principal argument is : Arg z   rad
4
Hence in polar form :

j   
z  2e 4
 2  cos  j sin 
 4 4

21. APPLICATIONS
 Complex numbers has a wide range of
applications in Science, Engineering,
Statistics etc.
Applied mathematics
Solving diff eqs with function of complex roots
 Cauchy's integral formula
 Calculus of residues
 In Electric circuits
to solve electric circuits

22. How complex numbers can be applied to
“The Real World”???
 Examples of the application of complex numbers:
1) Electric field and magnetic field.
2) Application in ohms law.
3) In the root locus method, it is especially important
whether the poles and zeros are in the left or right
half planes
4) A complex number could be used to represent the
position of an object in a two dimensional plane,

23. REFERENCES..
 Wikipedia.com
 Howstuffworks.com
 Advanced Engineering
Mathematics
 Complex Analysis

25. THANK YOU
FOR YOUR ATTENTION..!